A toaster uses a Nichrome heating wire. When the toaster is turned on at 20°C, the initial current is 1.50 A. A few seconds later, the toaster warms up and the current has a value of 1.20 A. The average temperature coefficient of resistivity for Nichrome wire is 4.5 10-4 (C°)-1. What is the temperature of the heating wire?

how do i do that

360

To find the temperature of the heating wire, we can use the average temperature coefficient of resistivity for Nichrome wire and the change in current.

The temperature coefficient of resistivity describes how the resistance of a material changes with temperature. It is given by the equation:

α = (1/R)(dR/dT)

where α is the temperature coefficient of resistivity, R is the resistance, and dT is the change in temperature.

In this case, we are given the initial and final current values, but we need to find the resistance at those temperatures. The resistance can be calculated using Ohm's law:

R = V/I

where R is the resistance, V is the voltage, and I is the current.

Since the voltage remains constant in this case, we can assume it cancels out when calculating the temperature coefficient of resistivity. So we can write:

α = (1/R₁)(dR₁/dT) = (1/R₂)(dR₂/dT)

where R₁ and R₂ are the resistances at the initial and final temperatures, respectively.

To calculate the change in resistance, we can use the formula:

ΔR = R₂ - R₁

Substituting these values into the equation for the temperature coefficient of resistivity, we get:

α = (1/R₁)(ΔR/dT) = (1/R₂)(ΔR/dT)

Now we can rearrange the equation to isolate the change in temperature:

ΔT = (ΔR/α)(1/R₁ + 1/R₂)

Plug in the known values:

ΔR = R₂ - R₁ = 1.20 A - 1.50 A = -0.30 A
α = 4.5 × 10^(-4) (C°)^(-1)
R₁ = V/I = (unknown voltage)/(1.50 A)
R₂ = V/I = (unknown voltage)/(1.20 A)

As we don't know the voltage, we need to solve for it by substituting the resistance values and rearranging the Ohm's law equation:

V = R₁ × I = (unknown voltage) × (1.50 A)
V = R₂ × I = (unknown voltage) × (1.20 A)

Equating the two expressions for voltage and solving for the unknown voltage:

R₁ × (1.50 A) = R₂ × (1.20 A)
R₁/R₂ = (1.20 A)/(1.50 A)

Now we can substitute the unknown voltage using the above equation:

V = (R₁/R₂) × (1.50 A)

Once we have the voltage, we can substitute it back into the equation for ΔT and solve for the change in temperature.

Finally, to find the temperature of the heating wire, add the change in temperature to the initial temperature of 20°C.

In steady state, the resistance is higher by a factor 5/4 because the current is lower by a factor 4/5.

Calculate the temperature rise necessary to increase the resistance by 25%. usng the temperature coefficient of resisitivity.

That will be the answer